A TWO-DIMENSIONAL REPRESENTATION OF FOUR-DIMENSIONAL GRAVITATIONAL WAVES

Abstract

The Einstein equation in D dimensions, if restricted to the class of spacetimes possessing n = D - 2 commuting hypersurface-orthogonal Killing vectors, can be equivalently written as metric-dilaton gravity in two dimensions with n scalar fields. For n = 2, this result reduces to the known reduction of certain four-dimensional metrics which include gravitational waves. Here, we give such a representation which leads to a new proof of the Birkhoff theorem for plane-symmetric spacetimes, and which leads to an explanation, in which sense two (spin zero-) scalar fields in two dimensions may incorporate the (spin two-) gravitational waves in four dimensions. (This result should not be mixed up with well–known analogous statements where, however, the four-dimensional spacetime is supposed to be spherically symmetric, and then, of course, the equivalent two-dimensional picture cannot mimic any gravitational waves.) Finally, remarks on hidden symmetries in two dimensions are made.